Here \(-\Delta _{g}\)  is the Laplace Beltrami operator and \(R_{g}\) 
is the scalar curvature of the original metric. In this case the new
metric \(\tilde{g}:=u^{\frac{4}{n-2}}g\)  has scalar curvature \(R_{\tilde{g}}\equiv \frac{4c(n-1)}{n-2}\) .
A positive answer to this problem was given by Yamabe [1]},
Trudinger [2]}, Aubin [3]} and Schoen [4]}.
